reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;
reserve p,p1,q,q1 for Element of Lambda(OASpace(V));
reserve POS for non empty ParOrtStr;
reserve p,p1,p2,q,q1,r,r1,r2 for Element of AMSpace(V,w,y);
reserve x,a,b,c,d,p,q,y for Element of POS;
reserve A,K,M for Subset of POS;
reserve POS for OrtAfSp;
reserve A,K,M,N for Subset of POS;
reserve a,b,c,d,p,q,r,s for Element of POS;

theorem Th58:
  b,c _|_ a,a & a,a _|_ b,c & b,c // a,a & a,a // b,c
proof
  reconsider a9=a,b9=b,c9=c as Element of the AffinStruct of POS;
  thus b,c _|_ a,a by Def7;
  hence a,a _|_ b,c by Def7;
  b9,c9 // a9,a9 & a9,a9 // b9,c9 by AFF_1:3;
  hence thesis by Th36;
end;
